Cell-impedance sensors

ABSTRACT

Systems, apparatus and methods for designing and operating a cell-electrode impedance sensor to detect chemical and biological samples, including biological cells. In one implementation, a method of designing a cell-electrode impedance sensor includes determining a cell free cell-electrode impedance and a cell covered cell-electrode impedance based on a design model for the cell-electrode impedance sensor, wherein the design model is based on one or more factors, the factors including properties and elements of a cell-electrode impedance measurement system, using the cell free cell-electrode impedance and the cell covered cell-electrode impedance to obtain a sensor sensitivity of the cell-electrode impedance measurement system, and choosing one or more design parameters of the cell-electrode impedance sensor in the cell-electrode impedance measurement system to maximize the sensor sensitivity.

CROSS REFERENCE TO RELATED APPLICATIONS

This patent document is a divisional application of and claims priorityto U.S. patent application Ser. No. 12/532,992, filed on Sep. 24, 2009,now U.S. Pat. No. 8,642,287, which is a U.S. National Stage under 35U.S.C. §371 of International Patent Application No. PCT/CN2007/002260,filed on Jul. 25, 2007, which claims benefit of priority of ChinesePatent Application No. 200710098717.4, filed on Apr. 25, 2007. Theentire contents of the before-mentioned patent applications areincorporated by reference as part of the disclosure of this application.

TECHNICAL FIELD

The present disclosure relates to detecting chemical and biologicalsamples, including biological cells.

BACKGROUND

Living cells are surrounded by an outer cell membrane that restricts themovement of ions and solutes between the cell interior and the exteriorof the cell. Changes in the electrical activity of cell membranes canreflect the biophysical state of the cells and further reflectunderlying physiological and biochemical processes occurring within thecell, as well as biophysical changes occurring at the surface of thecell or in the cell membrane. Electrical devices and apparatus for themeasurement of electrical impedance can be used to detect the state ofthe electrophysiological activity of living cells and their cellmembranes.

Cell-electrode impedance sensing and cell substrate electrical impedancesensing are two related electrical measurements that are based on theapplication of a small alternating current (AC) electrical signal toprobe the value of the impedance of sensor electrodes immersed in aconductive medium. Living cells can attach and grow on the surface ofthe sensor electrodes and can alter the electric field betweenelectrodes causing a change in the electrical impedance that can bedetected by the sensor electrodes. The measurement of impedance by thesensor can reflect the electrophysiological state of the cell and canallow the biophysical properties of the cell to be monitored.

SUMMARY

In one example, a method to design a cell-electrode impedancemeasurement system is described where a range of dimensions of acell-electrode impedance sensor in the system can be determined based ona model for cell-electrode impedance to maximize sensor sensitivity. Inaddition, a frequency range to optimize the parameters of the sensor(e.g., sensitivity, dimensions, etc.) can also be determined.

In one aspect, a method of designing a cell-electrode impedance sensoris described. The method includes determining a cell free cell-electrodeimpedance and a cell covered cell-electrode impedance based on a designmodel for the cell-electrode impedance sensor, wherein the design modelis based on one or more factors, the factors including properties andelements of a cell-electrode impedance measurement system, using thecell free cell-electrode impedance and the cell covered cell-electrodeimpedance to obtain a sensor sensitivity of the cell-electrode impedancemeasurement system, and choosing one or more design parameters of thecell-electrode impedance sensor in the cell-electrode impedancemeasurement system to maximize the sensor sensitivity.

This and other aspects can include one or more of the followingfeatures. The cell-electrode impedance sensor can include aninterdigitated electrode array comprising a plurality of branchelements. The design parameters can include one or more of a width ofthe branch elements, a length of the branch elements, a surface area ofthe branch elements, and a number of the branch elements. The method canfurther include choosing an effective width lesser than the width chosenbased on the design model, to account for a non-uniform electric fielddistribution over each branch element. The elements of thecell-electrode impedance measurement system can include one or more of acell culture vessel, a cell culture substrate, and circuitry. Theproperties of the cell-electrode impedance measurement system caninclude one or more of a property of a cell culture medium and aproperty of the cell. Determining the cell free cell-electrode impedanceand the cell covered cell-electrode impedance can include identifyingvalues for the properties of the cell-electrode impedance measurementsystem, substituting the values in a first equation associated with thecell free cell-electrode impedance to determine the cell freecell-electrode impedance, and substituting the values in a secondequation associated with the cell covered cell-electrode impedance todetermine the cell covered cell-electrode impedance. The cell freecell-electrode impedance can be represented by a Helmholtz interfacialcapacitance and a spreading resistance in the design model. Thespreading resistance can be associated with a proportionalitycoefficient. The cell covered cell-electrode impedance can berepresented by factors including one or more of a resistance between aplurality of cells, a capacitance between the plurality of cells, aresistance between a gap between a surface of a cell and a cell culturevessel substrate, and a capacitance between the surface of the cell andthe cell culture vessel substrate. The resistance and capacitancebetween the plurality of cells, and the resistance and capacitancebetween the gap between the surface of the cell and the cell culturevessel substrate can each be associated with a proportionalitycoefficient. The resistance and capacitance between the plurality ofcells, and the resistance and capacitance between the gap between thesurface of the cell and the cell culture vessel substrate can depend onan extent of attachment between a surface of the cells and a surface ofthe sensor. The surface of the cells can be completely attached to thesurface of the sensor. The surface of the cells can be partiallyattached to the surface of the sensor. Obtaining the sensor sensitivityof the system can further include determining the cell density of thesystem. The method can further include determining a frequency todeliver an alternating current (AC) signal to the sensor, whereindelivering the AC signal at the determined frequency can further improvesensor sensitivity. Determining the frequency can include determining aderivative of an equation for sensor sensitivity with respect tofrequency and solving for the frequency by equating the derivative tozero.

In another aspect, an apparatus for measuring cell-electrode impedancein biological cells is described. The apparatus includes a cell-culturevessel that includes electrodes and circuitry electrically coupled tothe electrodes. The electrodes are structured in an interdigitated arrayof electrode branch elements and the cell-culture vessel is operable tohold a cell-culture medium having biological cells attachable to theelectrodes. The circuitry includes a stimulator to apply an alternatingcurrent (AC) signal to the electrodes and configured to control afrequency of the AC signal to be between 10 kHz and 40 kHz.

This and other aspects can include one or more of the followingfeatures. The electrodes can be located in the cell-culture vessel at alocation to be immersed in the cell culture medium in the cell-culturevessel when the cell culture medium is present. The electrodes arrangedin an interdigitated array of electrode branch elements include firstelectrode branch elements connected to a first electrical terminal ofthe circuitry and second electrode branch elements connected to a secondelectrical terminal of the circuitry, wherein the first and secondelectrode branch elements are interleaved and the circuitry applies theAC signal at the first and second electrical terminals. Each branchelement can have a width. Adjacent branch elements can be separated by adistance. The branch element can have a width from 10 μm to 100 μm. Asurface of the cells can be attached to a surface of each branchelement. The cell-electrode impedance can depend on an extent of theattachment. A surface of the cells can be completely attached to thesurface of the electrodes. A surface of the cells can be partiallyattached to the surface of the electrodes. A width of the branchelements can be altered to account for the extent of the attachment. Thealtered width can equal the sum of a width of the branch element andtwice a radius of the biological cells.

In another aspect, a method for measuring cell-electrode impedance isdescribed. The method can include immersing an interdigitated electrodearray having electrode branch elements with a width from 10 μm to 100 μmin a cell culture medium in a cell-culture vessel to allow cells in thecell culture medium to attach to the electrode branch elements, andapplying an alternating current (AC) signal to the interdigitatedelectrode array in a frequency range from 10 kHz to 40 kHz to measure acell-electrode impedance for detecting cells.

This and other aspects can include one or more of the followingfeatures. The method can further include predetermining a number ofbranch elements, the branch element width, and a space between branchelements using a design model. The method can further include collectingan output from the electrode using a data acquisition module. The methodcan further include processing the collected output using a computersystem. The method can further include providing input to the electrodeusing the computer system. The frequency range can be chosen to maximizea sensitivity of the electrode when employed in a cell-electrodeimpedance measurement system.

The system and techniques described can provide one or more advantages.Firstly, the cell-electrode impedance measurement system can be designedto maximize sensitivity of the system and to maximize the number ofcells that can be studied. Further, a frequency of input AC signals canbe determined which can further maximize sensitivity of the system. Inaddition, cell-electrode impedance sensors can be designed based on adesign model.

The details of one or more implementations are set forth in theaccompanying drawings and the description below. Other features andadvantages will be apparent from the description and drawings, and fromthe claims.

DESCRIPTION OF DRAWINGS

FIG. 1 is an example of a schematic of a cell-electrode impedancemeasurement system.

FIG. 2 is an example of an equivalent circuit for cell freecell-electrode impedance.

FIG. 3A is an example of an equivalent circuit for cell coveredcell-electrode impedance.

FIG. 3B is an example of an equivalent circuit for cell coveredcell-electrode impedance when in low frequency range.

FIG. 3C is an example of an equivalent circuit for cell coveredcell-electrode impedance when in high frequency range.

FIG. 4 is an example of a cell-electrode impedance sensor.

FIG. 5A is an example of cell-electrode impedance sensor and anequivalent circuit for cell-electrode impedance.

FIG. 5B is an example of cell-electrode impedance sensor and anequivalent circuit for cell-electrode impedance.

FIG. 6 is an example of equivalent circuits of sensor electrodes withdifferent lengths.

FIG. 7 is a plot of sensor sensitivity in response to changingfrequency.

FIG. 8 is a plot of sensor sensitivity in response to changingfrequency.

FIG. 9 is a plot of sensor sensitivity in response to changingfrequency.

FIG. 10 is an example of a schematic of cells attached on an electrodesurface.

FIG. 11 is a plot validating the design model for cell-electrodeimpedance.

FIG. 12 is a plot of normalized impedance measured over time in responseto varied frequency.

FIG. 13 is a plot of sensor sensitivity in response to changingfrequency.

FIG. 14 is a flow chart of an example of designing a cell-electrodeimpedance measurement system.

FIG. 15 is a flow chart of an example of maximizing sensitivity of acell-electrode impedance measurement system.

Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION

FIG. 1 depicts an example of a cell-electrode impedance measurementsystem 100. The system 100 can include a cell culture vessel 105 andcircuitry 110. The cell culture vessel 105 can include elements toculture biological cells as well as elements to detect electricalproperties of the biological cells, such as electrodes. The circuitry110 can include elements to measure the electrical output produced bythe physiological changes undergone by the cells in the cell culturevessel 105. The circuitry 110 can include an AC stimulator 115, adetection resistor 120, a data acquisition module 125, and a digitalprocessor 130 (e.g., a computer). The AC stimulator can provide AC testsignals at different frequencies for cell-electrode impedance sensing.The data acquisition module 125 can be used to collect output signalsfrom the system 100. In some implementations, the output can include thevoltage between two electrodes in the system 100. In otherimplementations, the output can include voltages between one or morepairs of electrodes in the system. In addition, the output can includecurrent measurements, resistance measurements, etc. The rate ofcollection, for example, the number of voltage measurements per second,can be altered based on user input and the data acquisition module 125.The digital processor 130 can be used to store the data collected by thedata acquisition module 125, process the data and perform additionaloperations related to the processed data. For example, the digitalprocessor 130 may be configured to prepare spread sheets, performstatistical operations such as averaging, perform calculations such ascalculating impedance from measured voltages, currents, and resistances,etc., as well as transferring the data by methods including storing on aportable storage device (e.g., compact disc, USB flash drive, etc.),transmitting over a network (e.g., wired, wireless, the internet), etc.The digital processor 130 can also be configured to provide inputsignals to the system 100 to alter experimental conditions, e.g.,frequency of AC signals from the AC stimulator 115. In addition, thedigital processor 130 can be configured to receive input from a user.Such input can include constants for use in data processing, input toalter experimental conditions, etc.

The system 100 can include a cell culture vessel 105. The cell culturevessel 105 can include a vessel substrate 135, a cell culture medium140. Electrodes 145 can be positioned in the cell culture vessel 105.The electrodes 145 can be immersed in the cell culture medium 140. Insome implementations, the electrodes 145 can be arranged as aninterdigitated array and positioned such that a large surface area ofthe electrodes are available upon which the cells in the cell culturevessel can rest. The cell culture vessel 105 can contain the biologicalcells 105 under study. The cell culture vessel 105 can be manufacturedfrom a bio-compatible insulating material, e.g., plastic, glass,polydimethylsiloxane (PDMS), etc. The vessel substrate 135 can bemanufactured from the same material as the cell culture vessel 105. Thecell culture medium 140 can include proteins, sugar, salt, amino acids,antibiotics, and other nutrition that cells need. Cells 150 growing inthe cell culture vessel 105 can form a monolayer on the surface of thecell culture vessel 105 as well as the surface of the electrodes 145.Examples of cells that can be studied include HeLa, HepG2, NIN-3T3,VERO, etc. Properties of the cells that can be studied include cellproliferation, cell apoptosis, cell differentiation, cell migration,cell micromotion, cell attachment, etc. The cells 150 can be of severalshapes (e.g., spherical, hemispherical, etc.). A cell 150 can beattached either completely or partially to the surface of the electrode145. When a cell 150 is partially attached, only a portion of the cell150 can be in contact with the surface of the electrode 145. Inaddition, a cell can be attached to the surface of the cell culturevessel 105 and can have no contact with the surface of the electrode145.

In some implementations, the AC stimulator 115 can be designed andoperated to supply an input AC voltage at a pre-determined frequency.The frequency can be altered based on user input. The cell culturemedium 140 and the electrode 145 effectuate a part of a circuit throughwhich a probe current 155 is flowed. The presence of cells on theelectrode surface 145 can cause impedance which can affect the probecurrent 155. The probe current 155 can be affected by cells that areattached completely or partially attached to the surface of theelectrodes 145. The voltage between two electrode surfaces 145 can becollected by the data acquisition module 125 and stored in the digitalprocessor 130. Alternatively, or in addition, the probe current can becollected and stored in the digital processor 130. The digital processor130 can be configured to calculate the impedance due to the cells 150for every instance that the voltage and/or the probe current between theelectrode surfaces 145 were recorded. In this manner, the cell-impedanceof the system 100 can be determined.

The impedance of the system described in FIG. 1 can be attributed to atleast two sources, namely the cells 150 in the cell culture vessel 105and the cell culture vessel 105 itself. In the absence of cells 150, thesystem 100 is a cell-free cell-electrode impedance sensor. FIG. 2depicts an example of a schematic of an equivalent circuit for cell-freecell-electrode impedance for the system 100 in FIG. 1. The equivalentcircuit depicts representations of the Helmholtz double layerinterfacial capacitance, C_(I) (205) and the spreading resistance of thecell culture media, R_(S) (210).

FIG. 3A depicts an example of an equivalent circuit for cell coveredcell-electrode impedance for system 100. When cells attach and grow onthe electrode surface 145, the impedance of the cells 150 to the flow ofprobe current 155 can be modeled by the equivalent circuit shown in FIG.3A. The impedance can be a function of the electrical resistance betweengrowing cells, R_(cell) (310). In addition, the cells which can beattached on the electrode surface 145 can be treated as a capacitance,C_(cell) (320), which can be a biophysical property of the insulatingcell membrane. A gap can be present between a cell 150 growing on thesubstrate surface and the underside of the attached cell 150. The gapbetween the underside of the cell 150 and the surface of the substratecan be composed of the cell culture medium 140 and can cause aresistance, R_(gap) (305) and a capacitance, C_(gap)(315). The values ofR_(cell) (310), C_(cell)(320), R_(gap) (305), and C_(gap) (315) for acell 150 can depend on factors including one or more of the extent ofattachment of the cell 150 to the substrate surface, the biologicalnature of the cell, the cell culture medium 140, etc.

In some implementations, in measuring cell-electrode impedance, thesensitivity of the system 100 can be defined as the smallest impedance,caused by presence of cells in the cell culture vessel 105, that can bemeasured. The total impedance measured by the system 100 can be afunction of the impedance due to the cells 150, the impedance due to thecell culture medium 140, and the density of cells, Q_(cell). In someimplementations, the sensitivity of the system 100 can be a function ofthe cell free cell-electrode impedance (Z_(cell free-total)), the cellcovered cell-electrode impedance (Z_(cell-covered-total)), and the celldensity, Q_(cell), and can be depicted by equation (1).

$\begin{matrix}{{{Sensitivity}(f)} = \frac{{{Z_{{cell}\text{-}{covered}\text{-}{total}}(f)}} - {{Z_{{cell}\text{-}{free}\text{-}{total}}(f)}}}{Q_{cell}}} & (1)\end{matrix}$The impedance of the system 100 can be affected by the frequency, f, ofthe signal from the AC stimulator 115.

FIG. 3B depicts an equivalent circuit model for cell impedance when theapplied AC signal to the electrodes of the sensor is in a selected lowfrequency range. For example, this selected low frequency can rangebetween [(1/10)×(π·R_(cell)·C_(cell))] and [(1/10)×(π·R_(gap)·C_(gap))].In the low frequency range, the resistance of the cells, R_(cell) (310),and the resistance of the gaps, R_(gap) (305), dominate over thecapacitance of the cells, C_(cell) (320), and the capacitance of thegaps, C_(gap) (315).

FIG. 3C depicts an equivalent circuit model for cell impedance when theapplied AC signal to the electrodes of the sensor is in a selected highfrequency range. For example, this high frequency can range between[(5/2)×(π·R_(cell)·C_(cell)] and [(5/2)×(π·R_(gap)·C_(gap))]. In thehigh frequency range, the capacitance of the cells, C_(cell) (320), andthe capacitance of the gaps, C_(gap) (315), dominate over the resistanceof the cells, R_(cell) (310), and the resistance of the gaps, R_(gap)(305). The impedance of the system 100 can also be affected by thegeometry of the cell-electrode impedance sensor.

FIG. 4 depicts an example of a schematic of a cell-electrode impedancesensor 400. The sensor 400 can include two electrodes 403 and 405. Thetwo electrodes 403 and 405 can be configured to form an interdigitatedarray 410 formed of interleaved branch elements 415 where each branchelement 415 is connected to one sensor electrode 403 or 405 and theinterleaved branch elements 415 include one group of branch elements 415connected to the sensor electrode 403 and another group of branchelements 415 connected to the other sensor electrode 405. Referring toboth FIGS. 1 and 4, the first group of branch elements is connected to afirst electrical terminal of the circuitry 110 and the second group ofbranch elements is connected to a second electrical terminal of thecircuitry 110. The branch elements of the first and the second group canbe interleaved and the circuitry 110 applies the AC signal at the firstand second electrical terminals. In some implementations, the branchelements 415 of each electrode can be arranged in an alternating manner.Alternatively, the branch elements 415 can be arranged in any order, forexample, sequentially, alternatively, staggered, etc. In someimplementations, the branch elements 415 can also be positioned on bothsides of each electrode.

FIG. 4 also depicts a magnified view 420 of a portion of the sensor 400.The magnified view 420 illustrates branch elements 415 of the sensor. Inthe example illustrated in FIG. 4, the branch elements 415 of eachelectrode 405 are arranged in an alternating manner. In someimplementations, the branch elements 415 can have a width, W (425). Twoadjacent branch elements 15 can be separated by a distance, D (430). Insome implementations, the branch elements 415 of both electrodes 405 canhave a uniform width, W (425). In other implementations, the width, W(425), of the branch elements 415 of the same electrode 405 can beuniform. The width, W (425), of branch elements 415 of differentelectrodes can be non-uniform. In other implementations, the width, W(425), of each branch element 415 in the same electrode 405 can benon-uniform. In some implementations, the distance, D (430), between twobranch elements 415 can be uniform. In other implementations, the branchelements 415 can be staggered with a non-uniform distance, D (430),separating the branch elements 415.

The sensitivity of a sensor can depend on factors including the designparameters of the sensor 400. The design parameters of the sensor 400can include the material used to fabricate the sensor 400, the width, W(425), of each branch element 415, the distance between two branchelements, D (430), the length of the branch element 415, etc. The totalarea of the electrode can be represented by A_(total). The total lengthof the sensor 400 can be represented by L_(total). In someimplementations, the length of each branch element 415 can be uniform.In such implementations, the total length of the electrode, L_(total),can be represented by the length of each branch element, L, and thenumber of branch elements, N. In some implementations, the branchelement 415 can have a width between 10 μm and 100 μm, or between 10 μmand 80 μm, such as one of 20 μm, 40 μm, and 60 μm for a range of sensingapplications. The distance between two branch elements can be 20 μm insome sensor designs. The sensor 400 can be fabricated using variousmaterials such as glass, silicon, certain plastics, etc. by methodsincluding standard lift-off fabrication method. One method to determineoptimal design parameters of a sensor is to fabricate sensors ofdifferent design parameters and employ each sensor in a cell-electrodeimpedance measurement system.

In some implementations, a design model based on fundamental principlescan be developed to predict the cell free cell-electrode impedance andcell covered cell-electrode impedance of a cell-electrode impedancesystem 100. Based on the predicted impedance and given the conditions ofthe cell impedance measurement system 100, the design parameters of thesensor, such as branch element 415 dimensions, can be determined tooptimize the system 100 to increase sensitivity and increase number ofcells detected. In addition, the design model can be employed todetermine an optimal frequency to maximize sensitivity of the system100.

The total impedance measured by the system 100 can be a function of theimpedance due to the cells and the impedance due to the electrode. Theimpedance due to the electrode, represented by Z_(cell-free-total), canbe determined using equation (2).

$\begin{matrix}\begin{matrix}{{Z_{{cell}\text{-}{free}\text{-}{total}}(f)} = \frac{{2\; R_{S}} + {2( {j\; 2\;\pi\; f\; C_{I}} )^{- 1}}}{N}} \\{= \frac{{2\; R_{S}} + {2( {j\; 2\;\pi\; f\; C_{I}} )^{- 1}}}{{L_{total}/2}\; L}}\end{matrix} & (2)\end{matrix}$

When cells attach and grow on the surface of the sensor 400, theimpedance due to the cells, represented by Z_(cell-covered-total), canbe determined using equation (3).

$\begin{matrix}\begin{matrix}{{Z_{{cell}\text{-}{covered}\text{-}{total}}(f)} = \frac{\begin{matrix}{{2\; R_{S}} + {2( {j\; 2\;\pi\;{fC}_{I}} )^{- 1}} +} \\{2( {\frac{R_{cell} \times ( {j\; 2\;\pi\;{fC}_{cell}} )^{- 1}}{R_{cell} + ( {j\; 2\;\pi\;{fC}_{cell}} )^{- 1}} + \frac{R_{gap} \times ( {j\; 2\;\pi\;{fC}_{gap}} )^{- 1}}{R_{gapl} + ( {j\; 2\;\pi\;{fC}_{gap}} )^{- 1}}} )}\end{matrix}}{N}} \\{= \frac{\begin{matrix}{{2\; R_{S}} + {2( {j\; 2\;\pi\;{fC}_{I}} )^{- 1}} +} \\{2( {\frac{R_{cell} \times ( {j\; 2\;\pi\;{fC}_{cell}} )^{- 1}}{R_{cell} + ( {j\; 2\;\pi\;{fC}_{cell}} )^{- 1}} + \frac{R_{gap} \times ( {j\; 2\;\pi\;{fC}_{gap}} )^{- 1}}{R_{gapl} + ( {j\; 2\;\pi\;{fC}_{gap}} )^{- 1}}} )}\end{matrix}}{{L_{total}/2}\; L}}\end{matrix} & (3)\end{matrix}$

In the equivalent circuit equations (1) and (2), the interfacialcapacitance, C_(I) (205) can be a function of the permittivity of freespace, ∈₀, the effective dielectric constant of the double layerseparating the ionic charges and the electrode, ∈_(ρ), area of a singleelectrode branch element 415, A, and the thickness of the double layer,d_(dl). The interfacial capacitance, C_(I) (205) can be determined usingequation (4).

$\begin{matrix}{C_{I} = \frac{ɛ_{0}ɛ_{\rho}A}{d_{dl}}} & (4)\end{matrix}$

In the equivalent circuit equations (1) and (2), the spreadingresistance, R_(S) (210) can be a function of the resistivity of the cellculture medium 140, ρ, the area of a single electrode branch element415, A, and a proportionality coefficient, K. The spreading resistance,R_(S) (210) can be determined using equation (5) below:

$\begin{matrix}{R_{S} = \frac{\rho\; K}{\pi\; A}} & (5)\end{matrix}$

In the equivalent circuit equation (2), the resistance of the gapsbetween cells, R_(cell) 310, and the resistance of the cells to the gapbetween the cell and the substrate, R_(gap) 305, can be inverselyproportional to the electrode area, A. The capacitance of the cells thatare attached to the electrode surfaces, C_(cell) 320, and thecapacitance of the gap between the cell and the substrate, C_(gap) 315,can be directly proportional to the electrode area, A. The resistancesand the capacitances can be represented by equation (6) below:R _(cell)=K ₁ A ⁻¹; R _(gap)=K _(z) A ⁻¹; C _(cell)=K ₃ A; C _(gap)=K ₄A  (6)

The proportionality coefficients, K, K₁, K₂, K₃, and K₄ can bedetermined empirically. In one example, experimental data can be fittedto the design model using the least square fitting or other fittingalgorithm. In some implementations, the values of the proportionalitycoefficients can be 85 ohm·mm² and 48 ohm·mm² for K₁ and K₂,respectively, and 60 nF/mm² and 6 nF/mm² for K₃ and K₄, respectively.

Substituting equation (6) in equation (2) and equation (3), theimpedance due to the electrode, Z_(cell-free-total), and the impedancedue to the cells, Z_(cell-covered-total), can be represented by equation(7) and equation (8), respectively:

$\begin{matrix}\begin{matrix}{{Z_{{cell}\text{-}{free}\text{-}{total}}(f)} = \frac{2( {\frac{\rho\; K}{\pi} + ( {j\;\pi\; f\frac{ɛ_{0}ɛ_{\rho}}{d_{dl}}} )^{- 1}} )}{A.N}} \\{= \frac{2( {\frac{\rho\; K}{\pi} + ( {j\;\pi\; f\frac{ɛ_{0}ɛ_{\rho}}{d_{dl}}} )^{- 1}} )}{{{A.L_{total}}/2}\; L}}\end{matrix} & (7) \\\begin{matrix}{{Z_{{cell}\text{-}{covered}\text{-}{total}}(f)} = \frac{\begin{matrix}{{2( {\frac{\rho\; K}{\pi} + ( {j\;\pi\; f\frac{ɛ_{0}ɛ_{\rho}}{d_{dl}}} )^{- 1}} )} +} \\{2( {\frac{K_{1} \times ( {j\; 2\;\pi\;{fK}_{3}} )^{- 1}}{K_{1} \times ( {j\; 2\;\pi\;{fK}_{3}} )^{- 1}} + \frac{K_{2} \times ( {j\; 2\;\pi\;{fK}_{4}} )^{- 1}}{K_{2} + ( {j\; 2\;\pi\;{fK}_{4}} )^{- 1}}} )}\end{matrix}}{A.N}} \\{= \frac{\begin{matrix}{{2( {\frac{\rho\; K}{\pi} + ( {j\;\pi\; f\frac{ɛ_{0}ɛ_{\rho}}{d_{dl}}} )^{- 1}} )} +} \\{2( {\frac{K_{1} \times ( {j\; 2\;\pi\;{fK}_{3}} )^{- 1}}{K_{1} \times ( {j\; 2\;\pi\;{fK}_{3}} )^{- 1}} + \frac{K_{2} \times ( {j\; 2\;\pi\;{fK}_{4}} )^{- 1}}{K_{2} + ( {j\; 2\;\pi\;{fK}_{4}} )^{- 1}}} )}\end{matrix}}{{{A.L_{total}}/2}\; L}}\end{matrix} & (8)\end{matrix}$

According to equations (7) and (8), the cell free cell-electrodeimpedance and the cell covered cell-electrode impedance are inverselyproportional to the electrode area, A, and the total length of theelectrode, L_(total). The total length of the electrode, L_(total), canbe expressed as a function of the length of each branch element 415, L,and the number of branch elements, N. The total length of the electrode,L_(total), can be expressed by equation (9).L _(total)=2LN  (9)Equations (7) and (8) show that the cell free cell-impedance(Z_(cell-free-total)(f)) and the cell covered cell-impedance(Z_(cell-covered-total)(f)) are inversely proportional to the length ofeach branch element 415, L, and the number of branch elements, N. Fromequation (1), it can be determined that the sensitivity of thecell-impedance sensor is inversely proportional to the length of eachbranch element 415 and the number of branch elements. Therefore, in thedesign of a cell-electrode impedance sensor, the reduction of electrodebranch number or a reduction in the electrode width can improve thesensitivity of the sensor, but can decrease the number of cells whichcould be monitored by the sensor.

FIG. 5A and FIG. 5B show examples of cell-electrode impedance sensorsincluding branch elements of different areas. The cell-electrodeimpedance sensor depicted in FIG. 5A includes two branch elements 505and 510. The cell-electrode impedance sensor depicted in FIG. 5Bincludes two branch elements 520 and 525. In the examples shown, theareas of 505 and 510 are equal to each other while the areas of 520 and525 are equal to each other. The area of the branch elements 505 and 510is twice the area of the branch elements 520 and 525 in FIG. 5B. Theequivalent circuit of cell-electrode impedance in FIG. 5A and FIG. 5Bare represented by 515 and 530, respectively. In FIGS. 5A and 5B, theresistance represents the sum of R_(cell) (310) and R_(gap) (305) andthe capacitance represents the sum of C_(cell) (320) and C_(gap) (315).In the example shown, each branch element 505 and 510 contain 7 cellsattached to their surface. The equivalent circuit of each cellrepresents R_(cell) (310), R_(gap) (305), C_(cell) (320), and C_(gap)(315). Therefore, the total resistance and capacitance of cell culturevessel 505 is associated a factor of 1/7. In the example shown, thesurface of each electrode of cell culture vessel 510 contains twice thenumber of cells attached as the surface of electrodes of cell culturevessel 515. The equivalent circuit of each cell represents R_(cell)(310), R_(gap) (305), C_(cell) (320), and C_(gap) (315). Therefore, thetotal resistance and capacitance of cell culture vessel 510 isassociated a factor of 1/14. An electric field is generated between theelectrodes 505 and 510 and the electrodes 520 and 525 when a voltage isapplied.

FIG. 6 depicts an example of equivalent circuits of a cell-electrodeimpedance measurement systems of different lengths. The number of branchelements 415 in cell culture vessel 610 is twice the number of branchelements 415 in cell culture vessel 605. Thus, in the example shown, thetotal length of the electrodes in cell culture vessel 610 is twice thatof the electrodes in the cell culture vessel 605. The equivalent circuitof each cell represents R_(cell) (310), R_(gap) (305), C_(cell) (320),and C_(gap) (315). In the example shown, the surface of each electrodein both culture vessels contains the same number of cells attached.Since the number of branch elements 415 in cell culture vessel 610 istwice the number of branch elements 415 in cell culture vessel 605, thenumber of cells attached to the branch elements 415 of cell culturevessel 610 is twice the number of cells attached to the branch elements415 of cell culture vessel 605. Therefore, the total resistance andcapacitance of cell culture vessel 605 are associated with a factor of1/6. The total resistance and capacitance of cell culture vessel 610 areassociated with a factor of 1/12.

FIG. 7 depicts a plot of sensor sensitivity in response to frequency forthree different values for the total electrode length, L_(total). Thetotal electrode length, L_(total), is altered by increasing the numberof branch elements 415, N. The number of branch elements in the sensorstested were 40, 80, and 160. The width of the branch element, W, and theinter-electrode spacing, D, were maintained constant at 20 μm and 20 μm,respectively. As depicted in FIG. 7, the sensitivity (in ohms) increasedat the frequency range of 0-40 kHz and decreased for frequencies greaterthan 40 kHz.

FIG. 8 depicts a plot of sensor sensitivity in response to frequency fordifferent values for the inter-electrode distance, D. In one example,the number of branch elements 415 was 60 and the width of the electrode,W, was 20 μm. The inter-electrode spacing, D, was 20 μm, 40 μm, and 80μm. In a second example, the number of branch elements 415 was 60 andthe width of the electrode, W, was 40 μm. The inter-electrode spacing,D, was 20 μm, 40 μm, and 80 μm. In a third example, the number of branchelements 415 was 60 and the width of the electrode, W, was 80 μm. Theinter-electrode spacing, D, was 20 μm, 40 μm, and 80 μm. As depicted inFIG. 8, the sensitivity (in ohms) was not affected by the change ininter-electrode spacing, D.

FIG. 9 depicts a plot of sensor sensitivity in response to frequency fordifferent values of the electrode width, W. In one example, the numberof branch elements 415 was 60 and the inter-electrode distance, D, was20 μm. The width, W, was 20 μm, 40 μm, and 80 μm. In a second example,the number of branch elements was 60 and the inter-electrode distance,D, was 40 μm. The width, W, was 20 μm, 40 μm, and 80 μm. In a thirdexample, the number of branch elements 415 was 60 and theinter-electrode distance, D, was 80 μm. The width, W, was 20 μm, 40 μm,and 80 μm. As depicted in FIG. 7, the sensitivity (in ohms) decreasedwith increasing width, W.

FIG. 10 depicts an example of cells 150 totally or partially attached tothe surface of a branch element 415. In FIG. 7, cells 150, depicted by1, are positioned on the edge of the branch element 415 such that halfthe surface of the cell is attached to the branch element 415, whilehalf of the surface of the cell is not. Cells 150, depicted by 2, arepositioned on the edge of the branch element 415 such that more thanhalf of the surface of the cell is attached to the edge of the branchelement 415 while the remainder of the surface of the cell is not. Cells150, depicted by 3, are positioned on the edge of the branch element 415such that less than half of the surface of the cell is attached to theedge of the branch element 415, while the remainder of the cell is not.Cells 150, depicted by 4, are positioned on the branch element 415 suchthat the total surface of the cell is attached to the branch element415. Cells under category 4 can be positioned anywhere on the branchelement 415. In some implementations, only the cells that are attachedentirely to the branch element 415 can be used to determine cell coveredcell-electrode impedance. In other implementations, the contribution tothe measured signals from cells that are partially attached to thebranch elements 415 can be included.

Both, the cells attached totally and the cells attached partially, canbe defined as effective cells that contribute to the impedance. Numerouscells can occupy positions along the edges of the branch elements 415. Abranch element 415 can be represented by a length, L, and a width, W.The individual cells on the branch element 415 surface can fall underone of the categories 1, 2, 3, and 4 depicted in FIG. 7. The averagearea of any cell occupying a position on the edge of the branch element415 can be considered as equivalent to half the area of a cell 150. Thearea of a cell 150, in turn, can be represented as the distance of oneradius of a cell tangential to the long edge of the branch element 415for the total length of the edges of the branch elements 415. Aneffective branch element 415 width, W_(eff), can be defined by equation(10), to account for effective cells along both of the long edges of thebranch element 415.W _(eff) =W+2r _(cell)  (10)In addition, the effect of sharp turns of electrodes on the electricfield generated by the AC signal, known as the edge effect, can also beaccounted for by using the effective branch element width, W_(eff). Thesharp turns on the electrode can increase the strength of the localelectric field due to electric charge accumulation at the location ofthe sharp turn. This can alter the total electric field strength of theelectrode. The effect of the altered electric field near the sharp turnscan be accounted for by at least assuming the width of the branchelement, W, to be an effective width, W_(eff) which is set to be greaterthan the actual width W of the branch element. The relationshipregarding the effective electrode width and the effect of the electricfield can be complex when the electric field is a non-uniform fieldcaused by the edge effect where the strength of electric field in theedge is higher than that in the middle of the electrode. Experience andvarious tests have demonstrated that using a properly selected effectiveelectrode width W_(eff)>W can compensate for the non-uniformity of theelectric field distribution from the edge to the center of the electrodeduring the design process of the impedance sensors described in thisapplication. As a result of using this effective electrode width,W_(eff), the design model based on the uniform field distribution overthe electrode can be used to design actual sensors where the fielddistribution is not uniform due to the edge effect. An effective area ofthe branch element 415, A_(eff), can be defined and represented byequation (11):A _(eff) =W _(eff) ×L  (11)In some implementations, the length of the branch element can also bereplaced by an effective branch element length, which includes theradius of the cell near the short edge of the branch element 415.

The cell radius, r_(cell), is an intrinsic parameter of the type of theliving cell attached to the cell-electrode impedance sensor. Both, theedge effect and r_(cell), can affect the sensitivity of the sensor. Asthe width of the branch element 415 decreases, the cells which cover theedge of the branch element 415 can occupy a larger portion of thesurface are of the branch element 415. Both, A_(eff) and the edgeeffect, can be manipulated to improve the sensitivity of the sensor andcapturing the impedance from a larger number of cells as the width ofthe branch element 415 decreases.

The accuracy of equations (7) and (8), in predicting the cell free andcell covered cell-electrode impedance, respectively, can be determinedby experimentally measuring cell free and cell covered cell-electrodeimpedances in a cell-electrode impedance measurement system 100 andcomparing the measured values to the design prediction. For example,HeLa cell cultures with a cellular radius of about 9.5 μm were studied.Branch elements of widths, W, 20 μm, 40 μm, and 80 μm were chosen. Thisresulted in effective widths, W_(eff), of 39 μm (20+2*9.5), 59 μm(40+2*9.5), and 99 μm (80+2*9.5), respectively.

FIG. 11 depicts a comparison between cell-electrode impedance calculatedusing equations (7) and (8) and experimentally measured cell-electrodeimpedance using the HeLa cell cultures. The sensitivity of thecell-electrode impedance measurement system 100 is plotted against theeffective widths, W_(eff), of 3 branch elements 415. The experimentallymeasured sensitivity data, shown in FIG. 11, shows that the measuredsensitivity (represented by points in FIG. 11) lies on thecomputationally obtained fitted curve (represented by the line in FIG.11).

FIG. 12 depicts plots of normalized impedance measured over time inresponse to changing frequency. In the example shown, the cell-electrodeimpedance was measured during cell proliferation. FIG. 12 shows thatduring cell proliferation, as time increases, the frequency-dependentnormalized impedance increases (FIG. 12, (a)). It can also be seen fromFIG. 12 that, during cell proliferation, as time decreases, thefrequency-dependent normalized impedance decreases as time increases(FIG. 12, (b)).

FIG. 13 depicts the frequency characteristics of a cell-electrodeimpedance sensor. A frequency range to maximize the sensitivity of thecell-electrode impedance measurement system can be determined using thedesign model. In measuring cell-electrode impedance, AC signals from theAC stimulator 115 can be delivered at different frequencies. Threeimportant frequencies can be deduced from the equivalent circuitproposed in equation (7) and (8), namely, f_(low), f_(middle), andf_(high). In the low frequency range, interface capacitance, C_(I),dominates resistance and the equivalent impedance circuit can berepresented by FIG. 3B. The equivalent circuit which is effective in lowfrequency range can be a equated to a high-pass circuit and the cut-offfrequency at which the impedance of C_(I) is equal to the sum of R_(S),R_(cell), and R_(gap) can be defined by equation (12).

$\begin{matrix}{f_{{cut}\text{-}{off}\text{-}{low}} = \frac{1}{2\;{\pi( {R_{S} + R_{cell} + R_{gap}} )}C_{I}}} & (12)\end{matrix}$A frequency, f_(low), can be defined as lower than one fifth one-fifthof f_(cut-off-low). At a frequency lower than f_(low), the impedance ofcell covered cell-electrode sensor can be dominated by interfacecapacitance, C_(I). In this range, the impedance of cell freecell-electrode sensor is also dominated by C_(I) and the sensitivity ofthe sensor can be low. Further, as the frequency decreases, thesensitivity can decrease and approach zero. The frequency, f_(low), canbe represented by equation (13).

$\begin{matrix}\begin{matrix}{f_{low} = \frac{f_{{cut}\text{-}{off}\text{-}{low}}}{5}} \\{= \frac{1}{10\;{\pi( {R_{S} + R_{cell} + R_{gap}} )}C_{I}}} \\{= \frac{1}{10\;{\pi( {\frac{\rho\; K}{\pi\; A} + {K_{1}A^{- 1}} + {K_{2}A^{- 1}}} )}\frac{ɛ_{0}ɛ_{\rho}A}{d_{dl}}}} \\{= \frac{1}{10\;{\pi( {( {\frac{\rho\; K}{\pi} + K_{1} + K_{2}} )\frac{ɛ_{0}ɛ_{\rho}}{d_{dl}}} )}}}\end{matrix} & (13)\end{matrix}$

In the high frequency range, R_(cell) can dominate C_(cell), and R_(gap)can dominate C_(gap). The equivalent impedance circuit can berepresented by FIG. 3C. The equivalent circuit can be equated to ahigh-pass circuit and the cut off frequency can be represented byequation (14).

$\begin{matrix}{f_{{cut}\text{-}{off}\text{-}{high}} = \frac{1}{2\;\pi\;{R_{S}( {C_{I}^{- 1} + C_{cell}^{- 1} + C_{gap}^{- 1}} )}}} & (14)\end{matrix}$A frequency, f_(high), can be defined as five times f_(cut-off-high). Ata frequency higher than f_(cut-off-high), the impedance of both cellfree and cell covered cell-electrode impedance can be dominated byR_(S). Further, the sensitivity can be low and can approach zero as thefrequency increases. The frequency, f_(high), can be represented byequation (15).

$\begin{matrix}\begin{matrix}{f_{high} = {5 \times f_{{cut}\text{-}{off}\text{-}{high}}}} \\{= \frac{5}{2\;\pi\;{R_{S}( {C_{I}^{- 1} + C_{cell}^{- 1} + C_{gap}^{- 1}} )}}} \\{= \frac{5}{2\;\pi\frac{\rho\; K}{\pi\; A}( {\frac{d_{dl}}{ɛ_{0}ɛ_{\rho}A} + \frac{1}{K_{3}A} + \frac{1}{K_{4}A}} )^{- 1}}} \\{= \frac{1}{2\;\pi\frac{\rho\; K}{\pi}( {\frac{d_{dl}}{ɛ_{0}ɛ_{\rho}} + \frac{1}{K_{3}} + \frac{1}{K_{4}}} )^{- 1}}}\end{matrix} & (15)\end{matrix}$

A third frequency, f_(middle), can be defined as the frequency at whichthe sensitivity can be maximum. This frequency can be determined bysolving the derivative of the equation for sensitivity (1). Thederivative can be represented by equation (16).

$\begin{matrix}\begin{matrix}{\frac{\mathbb{d}( {{Sensitivity}(f)} )}{\mathbb{d}f} = \frac{\mathbb{d}( {{{Z_{{cell}\text{-}{covered}\text{-}{total}}(f)}} - {{Z_{{cell}\text{-}{free}\text{-}{total}}(f)}}} )}{\mathbb{d}f}} \\{= 0}\end{matrix} & (16)\end{matrix}$As illustrated in FIG. 13(a), the cell free cell-electrode impedance andthe cell covered cell-electrode impedance are nearly equal atfrequencies closer to f_(low) and f_(high). The difference between thecell free cell-electrode impedance and the cell covered cell-coveredimpedance is greatest at frequencies near f_(middle). Since thesensitivity is directly proportional to the difference between the cellcovered cell-electrode impedance and the cell free cell-electrodeimpedance, sensitivity of the impedance measurement system can bemaximized at and near a frequency equal to f_(middle). The dimensions ofa sensor are independent of the frequency of the AC signal. In someimplementations, the sensitivity can be maximized when the frequencyrange of the input signal is between 10 kHz and 40 kHz. This operatingfrequency range from 10 kHz to 40 kHz can be determined based onexperimental data of the sensors. Equivalent circuit parametersdescribed in this specification can be used to fit to measuredexperimental data to determine equivalent circuit parameters and theresultant equivalent circuit parameters are then applied to the circuitequations to calculate the operating frequency range for the sensor.

FIG. 14 depicts a flow chart of an example of designing a cell-electrodeimpedance measurement system. The values of parameters of the system canbe determined at 1405. The parameters can include one or more of theproperties of the cell culture vessel, the culture medium, the culturecells, the proportion coefficients to represent R_(gap), R_(cell),C_(gap), and C_(cell), etc. The cell free cell-electrode impedance canbe determined from a design model at 1410. The cell free cell-electrodeimpedance can be determined using equation (7). The cell coveredcell-electrode impedance can be determined from the design model at1415. The computational cell covered cell-electrode impedance can bedetermined using equation (8). Using the computationally determined cellfree and cell covered cell-electrode impedances, the sensitivity of asystem can also be determined at 1420. The computational sensitivity canbe determined using equation (1). The computational cell free and cellcovered cell-electrode impedances can be used to predict parameters of acell-electrode impedance sensor in the system at 1425. Thecell-electrode impedance sensor can be an interdigitated array. Thepredicted parameters can include the area of the branch elements of theinterdigitated array, the width of the branch elements, and the numberof branch elements. Based on equations (7) and (8), the sensitivity ofthe sensor is inversely proportional to the area of the branch elementsand the number of branch elements. In addition, the area of the branchelements and the number of elements is directly proportional to thenumber of cells that can be studied. By predicting the cell-electrodeimpedances, a sensor can be designed that can maximize sensitivity whilealso maximizing the number of cells that can be studied.

FIG. 15 depicts a flow chart of an example of optimizing the frequencyof AC signals applied to a cell-electrode impedance measurement system.Based on the sensitivity of the cell-electrode impedance measurementsystem, determined from equation (1), and the cell free and cell coveredcell-electrode impedances determined from equations (7) and (8), theoptimum frequency of AC signals can be determined at 1505. In someimplementations, the equations representing cell free cell-electrodeimpedance (7) and cell covered cell-electrode impedance (8) can besubstituted into the equation representing sensitivity equation (1). Theequation representing sensitivity can be differentiated with respect tofrequency and the resulting expression can be set to zero. A solution tothe equation can yield an optimum frequency value. AC signals can beapplied to the cell-electrode impedance measurement system at thedetermined optimum frequencies at 1510. The cell free and cell coveredcell-electrode impedances of the system can be measured at 1515.

Hence, one design model for designing a cell-electrode impedance sensorsystem is based on an equivalent circuit model for interdigitated arraystructure cell-electrode impedance sensor, including a first equivalentcircuit model the cell free sensor and a second equivalent circuit modelfor the cell covered sensor. The equivalent circuit model in this designmodel can include parameters Rcell, Ccell, Rgap and Cgap. Rcell is theresistance of the mean gap distance between the cells which are attachedon the electrodes of the sensor. This parameter can be used to monitorthe mean gap distance between cells in real time. Ccell is thecapacitance of cells created by their insulating cell membranes. Thisparameter can reflect changes in the biophysical properties of the cellmembranes. Rgap is the resistance of the gap between the attached cellsand the electrode surfaces. This parameter can be used in real timemonitor the mean gap between the attached cells and electrode surfacesubstrate. Cgap is the capacitance between the attached cells and theelectrode surface substrate. This parameter can be used to reflect thegap between attached cell membranes and electrode surface substrate.

Part or entirety of the design processes described in this specificationmay be implemented as computer-implemented processes. The designparameters obtained from the described design processes can be used tofabricate sensor devices in a wide range of applications includingsensing biological cells.

While this specification contains many specifics, these should not beconstrued as limitations on the scope of any invention or of what may beclaimed, but rather as descriptions of features specific to particularembodiments. Certain features that are described in this specificationin the context of separate embodiments can also be implemented incombination in a single embodiment. Conversely, various features thatare described in the context of a single embodiment can also beimplemented in multiple embodiments separately or in any suitablesubcombination. Moreover, although features may be described above asacting in certain combinations and even initially claimed as such, oneor more features from a claimed combination can in some cases be excisedfrom the combination, and the claimed combination may be directed to asubcombination or variation of a subcombination. Only a fewimplementations and examples are described. However, other variations,modifications, and enhancements are possible and are within the scope ofthe following claims.

What is claimed is:
 1. An apparatus for measuring cell-electrodeimpedance in biological cells, comprising: a cell-culture vesselcomprising electrodes structured in an interdigitated array of electrodebranch elements, the cell-culture vessel operable to hold a cell-culturemedium having biological cells attachable to the electrodes and theelectrode branch elements arranged to be staggered with a non-uniformdistance between each of the electrode branch elements; and circuitryelectrically coupled to the electrodes and comprising a detectionresistor and a stimulator to apply an alternating current (AC) signal atdifferent frequencies to the electrodes and configured to control afrequency range of the AC signal to be between a low frequency and ahigh frequency; wherein the electrodes have an equivalent electricalcircuit that is dominated by interface capacitance at the low frequencyand is dominated by spreading resistance at the high frequency.
 2. Theapparatus of claim 1, wherein the electrodes are located in thecell-culture vessel at a location to be immersed in the cell culturemedium in the cell-culture vessel when the cell culture medium ispresent.
 3. The apparatus of claim 1, wherein the electrode branchelements include first electrode branch elements connected to a firstelectrical terminal of the circuitry and second electrode branchelements connected to a second electrical terminal of the circuitry,wherein the first and second electrode branch elements are interleavedand the circuitry applies the AC signal at the first and secondelectrical terminals.
 4. The apparatus of claim 1, wherein each branchelement has a width from 10 μm to 100 μm.
 5. The apparatus of claim 1,wherein the cell-culture vessel comprises a bio-compatible insulatingmaterial.
 6. The apparatus of claim 5, further comprising a vesselsubstrate that is manufactured from a same material as the cell-culturevessel.
 7. The apparatus of claim 1, further including: a digital signalprocessor that collects and stores values of at least one of the ACsignal and a voltage difference between electrodes.
 8. The apparatus ofclaim 1, wherein the interdigitated array of electrode branch elementsincludes a first electrode comprising interleaved branch elements and asecond electrode comprising second interleaved branch elements.
 9. Theapparatus of claim 1, wherein the sensitivity of the electrodescorresponds to smallest impedance caused by presence of cells in thecell-culture vessel that can be measured by the apparatus.
 10. Theapparatus of claim 1, wherein the sensitivity is a function of acell-free impedance of the electrodes, a cell-covered impedance of theelectrodes and a density of the biological cells.
 11. The apparatus ofclaim 1, wherein the sensitivity of the electrodes is a function offrequency of the AC signal.
 12. A cell-electrode impedance sensor foruse in a cell-electrode impedance measurement system, comprising: aninterdigitated electrode array comprising a plurality of branch elementsthat have non-uniform actual widths and are separated by non-uniformdistances, the plurality of branch elements characterized by designparameters including one or more effective widths of the branchelements, a length of the branch elements, a surface area of the branchelements, and a number of the branch elements; wherein the designparameters are determined based on a design model for the cell-electrodeimpedance sensor, the design model predicting a cell free cell-electrodeimpedance and a cell covered cell-electrode impedance; and wherein eachof the non-uniform actual widths of the branch elements is less thaneach of the effective width of the branch elements obtained based on thedesign parameters, to account for a non-uniform electric fielddistribution over each branch element; wherein the interdigitatedelectrode array has an equivalent electrical circuit that is dominatedby interface capacitance at a low frequency and is dominated byspreading resistance at a high frequency.
 13. The cell-electrodeimpedance sensor of claim 12, wherein the cell-electrode impedancemeasurement system includes one or more of a cell culture vessel, a cellculture substrate, and circuitry.
 14. The cell-electrode impedancesensor of claim 12, wherein the cell-electrode impedance sensor isfabricated using various materials such as glass, silicon and plastics.15. The cell-electrode impedance sensor of claim 12, wherein thecell-electrode impedance measurement system includes: a digital signalprocessor that collects and stores values of at least one a currentsignal and a voltage difference signal of the cell-electrode impedancesensor.
 16. The cell-electrode impedance sensor of claim 12, wherein theinterdigitated electrode array includes a first electrode comprisinginterleaved branch elements and a second electrode comprising secondinterleaved branch elements.
 17. The cell-electrode impedance sensor ofclaim 12, wherein the sensor sensitivity corresponds to smallestimpedance caused by presence of cells in the cell-culture vessel thatcan be measured by the apparatus.